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Inmathematics, thetensor bundle of amanifold is thedirect sum of all tensor products of thetangent bundle and thecotangent bundle of that manifold. To docalculus on the tensor bundle aconnection is needed, except for the special case of theexterior derivative of antisymmetric tensors.

A tensor bundle is afiber bundle where the fiber is a tensor product of any number of copies of thetangent space and/orcotangent space of the base space, which is a manifold. As such, the fiber is avector space and the tensor bundle is a special kind ofvector bundle.

• Lee, John M. (2012).Introduction to Smooth Manifolds.Graduate Texts in Mathematics. Vol. 218 (Second ed.). New York London:Springer-Verlag.ISBN 978-1-4419-9981-8.OCLC 808682771.
• Saunders, David J. (1989).The Geometry of Jet Bundles. London Mathematical Society Lecture Note Series. Vol. 142. Cambridge New York:Cambridge University Press.ISBN 978-0-521-36948-0.OCLC 839304386.
• Steenrod, Norman (5 April 1999).The Topology of Fibre Bundles. Princeton Mathematical Series. Vol. 14. Princeton, N.J.: Princeton University Press.ISBN 978-0-691-00548-5.OCLC 40734875.

• Fiber bundle – Continuous surjection satisfying a local triviality condition
• Spinor bundle – Geometric structure
• Tensor field – Assignment of a tensor continuously varying across a region of space

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